GMRES and the Arioli,Pták,and Strakoš parametrization

In this paper we study the parametrization proposed by Arioli, Pták and Strakoš (BIT Numerical Mathematics, v 38, 1998) for the class of matrices having the same GMRES residual norm convergence curve. We give expressions for the Hessenberg matrix and the orthonormal basis vectors constructed by GMRES as well as for iterates and error vectors. The iterates do not depend on the eigenvalues in the sense that changing the coefficients of the characteristic polynomial in the parametrization does not change the GMRES iterates as well as the residuals. However, the error vectors do depend on these coefficients.     

Preferences,agreement, consensus — measuring,aggregation and control

This paper presents a model for analysis and a practical method of management of a session meant at attainment of consensus regarding preferences over a set of multi-aspect options, these preferences being expressed through pairwise comparisons or orderings. The model and the method accept and process valued or fuzzy preferences, while preserving an entirely crisp procedure throughout the approach. The aggregation method presented previously in Owsiski and Zadrony [17, 18] is referred to and expanded for the case of clusterwise preference aggregation and consensus measurement. The framework of the clustering approach of Owsiski [19] is applied, allowing for a natural definition of consensus.     

Justification,Justifying, and Leite’s Localism

In a series of papers, Adam Leite has developed a novel view of justification tied to being able to responsibly justify a belief. Leite touts his view as (i) faithful to our ordinary practice of justifying beliefs, (ii) providing a novel response to an epistemological problem of the infinite regress, and (iii) resolving the “persistent interlocutor” problem. Though I find elements of Leite’s view of being able to justify a belief promising, I hold that there are several problems afflicting the overall picture of justification. In this paper, I argue that despite its ambitions, Leite’s view fails to solve the persistent interlocutor problem and does not avoid a vicious regress.     

Riemannian submersions, δ-invariants, and optimal inequality

In this study, we establish a sharp relation between δ-invariants and Riemannian submersions with totally geodesic fibers. By using this relationship, we establish an optimal inequality involving δ-invariants for submanifolds of the complex projective space CP ^m (4) via Hopf’s fibration ${\pi:S^{2m+1}\to CP^{m}(4)}$ . Moreover, we completely classify submanifolds of complex projective space which satisfy the equality case of the inequality.     

Euclid,Fibonacci, and Pascal—recursed!

Selected algorithms of three famous mathematicians are expressed in recursive computer programmes using a programming language (APL). Specifically: (a) Euclid's greatest common divisior algorithm; (b) Fibonacci's ‘ golden ‘ number series; (c) Pascal's triangle.

The programmes are designed to be viewed directly by students in order to elucidate the algorithms. Such programmes can be stimulants for learning mathematics.     

On roots and squares – estimation,intuition and creativity

The paper presents findings of a small scale study of a few items related to problem solving with squares and roots, for different teacher groups (pre-service and in-service mathematics teachers: elementary and junior high school). The research participants were asked to explain what would be the units digit of a natural number to be squared in order to obtain a certain units digit as a result. They were also asked to formulate a rule – an algorithm for calculating the square of a 2-digit number which is a multiple of 5. Based on this knowledge and estimation capability, they were required to find, without using calculators, the square roots of given natural numbers. The findings show that most of the participants had only partial intuition regarding the units’ digit of a number which is squared when the units’ digit of the square is known. At the same time, the participants manifested some evidence of creativity and flow of ideas in identifying the rule for calculating the square of a natural number whose units digit is 5. However, when asked to identify, by means of estimation and based on knowledge from previous items, the square roots of three natural numbers, only few of them managed to find the three roots by estimation.     

On non-isospectral flows,Painlevé equations,and symmetries of differential and difference equations

We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called discrete Painlevé equations cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide delay Painlevé equations.In Memory of Prof. M. C. PolivanovDipartimento di Fisica, P. le A. Moro 2, 00185 Roma, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy. Departamento de Fisica Teorica, Facultad de Fisicas, Universidad Complutense, 28040 Madrid, Spain. W561@emducm11.bitnet. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 473–480, December, 1992.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Projective Valuations,D-Relations,and Ends of Λ-Trees

Abstract

A projective valuation on a set Eis a mapping w : E ⁴ → Λ ∪ {±∞}, where Λ is an ordered abelian group, satisfying certain axioms. A D-relation on Eis a four-place relation on E, again with certain properties. There is a projective valuation on the set of ends of a Λ-tree (and on any subset, by restriction) and we show, using a construction suggested by Tits in the case Λ = ?, that every projective valuation arises in this way. Every projective valuation wdefines a D-relation, and there is a simple geometric interpretation of the D-relation, given a Λ-tree defining w. Our main result is a converse, that any D-relation can be defined by a projective valuation, hence arises from an embedding into the set of ends of a Λ-tree.     

Absorbing Angles,Steiner Minimal Trees,and Antipodality

We give a new proof that a star {op _i :i=1,…,k} in a normed plane is a Steiner minimal tree of vertices {o,p ₁,…,p _k } if and only if all angles formed by the edges at o are absorbing (Swanepoel in Networks 36: 104–113, 2000). The proof is simpler and yet more conceptual than the original one. We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star {op _i :i=1,…,k} in any CL-space is a Steiner minimal tree of vertices {o,p ₁,…,p _k } if and only if all angles are absorbing, which in turn holds if and only if all distances between the normalizations \frac1||p_i||p_i\frac{1}{\Vert p_{i}\Vert}p_{i} equal 2. CL-spaces include the mixed ℓ ₁ and ℓ _∞ sum of finitely many copies of ℝ.     

Foliations, Groupoids, and the Baum―Connes Conjecture

The BaumConnes conjecture establishes, for foliated manifolds, an analog of the well-known isomorphism between the topological K-theory of a locally compact space M and the analytic K-theory of the C ^*-algebra of continuous functions on M vanishing at infinity. In this work, we describe the principal notions involved in the statement of the conjecture and indicate its contemporary status. Bibliography: 11 titles.     

Duality,area-considerations,and the Kalai–Smorodinsky solution

We introduce a new solution concept for $2$-person bargaining problems, which can be considered as the dual of the Equal-Area solution (EA) (see Anbarc? and Bigelow (1994)). Hence, we call it the Dual Equal-Area solution (DEA). We show that the point selected by the Kalai–Smorodinsky solution (see Kalai and Smorodinsky (1975)) lies in between those that are selected by EA and DEA. We formulate an axiom–area-based fairness–and offer three characterizations of the Kalai–Smorodinsky solution in which this axiom plays a central role.     

Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the -system labels its decreasing chains.Basic cycles can be carried over from The intersection poset of any (real or complex) afflnehyperplane arrangement is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc(M), for the union of such an arrangement.     

Relative Flatness,Mittag–Leffler Modules,and Endocoherence

Let M _R be a right R-module over a ring R with S = End(M _R ). We study the coherence of the left S-module _S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.     

Mathias–Prikry and Laver type forcing; summable ideals,coideals, and +-selective filters

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.     

Dimensional analysis,modelling and symmetry†

In this paper the author discusses the controversial question of the future of ‘ new ‘ mathematics.